Podcast
Questions and Answers
What is the value of $x$ in the equation $3x + 7 = 16$?
What is the value of $x$ in the equation $3x + 7 = 16$?
- 4
- 5 (correct)
- 3
- 2
If the area of a triangle is given by the formula $A = \frac{1}{2}bh$, what is the area when the base $b$ is 10 and the height $h$ is 5?
If the area of a triangle is given by the formula $A = \frac{1}{2}bh$, what is the area when the base $b$ is 10 and the height $h$ is 5?
- 25 (correct)
- 100
- 50
- 30
What is the slope of the line represented by the equation $y = 4x + 2$?
What is the slope of the line represented by the equation $y = 4x + 2$?
- 0
- -4
- 2
- 4 (correct)
If the roots of the quadratic equation $x^2 - 5x + 6 = 0$ are $p$ and $q$, what is the value of $pq$?
If the roots of the quadratic equation $x^2 - 5x + 6 = 0$ are $p$ and $q$, what is the value of $pq$?
What is the derivative of the function $f(x) = x^3 - 3x^2 + 4$?
What is the derivative of the function $f(x) = x^3 - 3x^2 + 4$?
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Study Notes
Solving Linear Equations
- The equation $3x + 7 = 16$ can be solved for $x$ by isolating the variable.
- Subtract 7 from both sides to obtain $3x = 9$.
- Divide both sides by 3 to get $x = 3$.
Calculating Triangle Area
- The formula for the area of a triangle is $A = \frac{1}{2}bh$, where $b$ is the base and $h$ is the height.
- Substitute $b = 10$ and $h = 5$ into the formula: $A = \frac{1}{2}(10)(5)$.
- Simplify to get $A = 25$.
Identifying Slope
- The equation $y = 4x + 2$ is in slope-intercept form ($y = mx + c$), where $m$ represents the slope.
- Therefore, the slope of the line is $m = 4$.
Understanding Quadratic Roots
- The product of the roots of a quadratic equation in the form $ax^2 + bx + c = 0$ is given by $c/a$.
- In the equation $x^2 - 5x + 6 = 0$, the product of the roots $p$ and $q$ is $(6)/(1) = 6$.
Finding Derivatives
- The derivative of a function $f(x) = x^n$ is given by $f'(x) = nx^{n-1}$.
- Applying this to the function $f(x) = x^3 - 3x^2 + 4$, we get the derivative $f'(x) = 3x^2 - 6x$.
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